COMPUTER NUMERICAL CONTROL
OF MACHINE TOOLS
Laboratory for Manufacturing Systems
Department of Mechanical Engineering and Automation
University of Patras, Greece
Laboratory for Manufacturing Systems and Automation
Dr.Dimitris Mourtzis
Chapter 8:
Mathematics for Numerical Control
Programming
Laboratory for Manufacturing Systems and Automation
Dr.Dimitris Mourtzis
Objectives of Chapter 8
Use right
right--angle trigonometry to determine programming
coordinates from part drawings
Laboratory for Manufacturing Systems and Automation
Dr.Dimitris Mourtzis
Basic Trigonometry
FIGURE 88-1 Right triangle
Laboratory for Manufacturing Systems and Automation
Dr.Dimitris Mourtzis
Basic Trigonometry
OPPOSITE SIDE
SIE A =
HYPOTEUSE
a
SIE 30 =
1.500
a
0.500 =
1.500
a = 0.750
FIGURE 88-2
Laboratory for Manufacturing Systems and Automation
Dr.Dimitris Mourtzis
Basic Trigonometry
ADJACET SIDE
COSIE A =
OPPOSITE SIDE
TA 40 =
HYPOTEUSE
ADJACETSIDE
1.000
b
TA 40 =
COS A =
b
1.500
1.000
b
0.839 =
0.866 =
1.500
b = 1.299
Laboratory for Manufacturing Systems and Automation
b
b = 1.191
Dr.Dimitris Mourtzis
Basic Trigonometry
Dimension X equals 3.000 – 1.191
191,, or 1.809
FIGURE 88-3
Laboratory for Manufacturing Systems and Automation
Dr.Dimitris Mourtzis
Using Trigonometry for Cutter Offsets
1.
2.
3.
The total number of degrees in a
circle is 360
The sum of the angles in a triangle
is 180 degrees
The complement of an angle is 90
minus the angle
FIGURE 88-4
Laboratory for Manufacturing Systems and Automation
Dr.Dimitris Mourtzis
Using Trigonometry for Cutter Offsets
Solving triangle Y for ∆Υ
∆Υ::
∆Υ
= ΤΑΝ 25
0.250
(
∆Υ = ΤΑΝ 25 0.250
)
∆Υ = 0.11658 or 0.117
Solving triangle Χ for ∆Χ
∆Χ::
∆Χ
= ΤΑΝ 20
0.250
(
∆Χ = ΤΑΝ 20 0.250
FIGURE 88-5
Laboratory for Manufacturing Systems and Automation
)
∆Χ = 0.09099 or 0.091
Dr.Dimitris Mourtzis
Using Trigonometry for Cutter Offsets
FIGURE 88-6 Solutions of right triangles
Laboratory for Manufacturing Systems and Automation
Dr.Dimitris Mourtzis
Using Trigonometry for Cutter Offsets
FIGURE 88-7 Solutions of oblique
oblique--angled triangles
Laboratory for Manufacturing Systems and Automation
Dr.Dimitris Mourtzis
A Milling Example
0.185
SI θ =
or 0.21143
0.875
θ = 12.206 degrees
Once angle θ is known, angle α and
angle β can be determined
determined::
α = 30° − θ and β = 60° − θ
The Pythagorean formula is used to
determine the Y coordinate of P1:
FIGURE 88-8
Χ = 0.185 ( known informatio n)
Y =
Laboratory for Manufacturing Systems and Automation
2
0.875 − 0.185
2
or 0.85522
Dr.Dimitris Mourtzis
A Milling Example
The coordinates of P2 are calculated by the SIN and COS of angle α:
P2 :
X = 0.875 ( COSα ) or 0.83314
[
]
Y = − 0.875( SIα ) or − 0.2674
The coordinates of P3
P3 are calculated by the SIN and COS of angle β:
X = 0.875 ( COSβ ) or 0.5878
[
]
Y = − 0.875( SIβ ) or − 0.6481
Laboratory for Manufacturing Systems and Automation
Dr.Dimitris Mourtzis
A Lathe Example
FIGURE 88-9
Laboratory for Manufacturing Systems and Automation
Dr.Dimitris Mourtzis
A Lathe Example
FIGURE 88-10
Laboratory for Manufacturing Systems and Automation
Dr.Dimitris Mourtzis
A Lathe Example
To determine the preliminary information:
Α = 0.250( ΤΑΝ 37 ) or 0.18839
∆Κ = 0.005( ΤΑΝ 18.5 ) or 0.00167
∆Ι = 0.005( ΤΑΝ 26.5 ) or 0.00249
To determine the cutter locations:
P1 :
Χ = 0.750 − Α − ∆Ι
Χ = 0.55912
Ζ = − 0.005 + 0.020
Ζ = 0.015
P2 :
Note: 0.55912 in X formula from P1 X calculation
Χ = 0.55912 + 0.020( COS 37° )
Z = − 0.005 + 0.02( SI 37° )
Χ = 0.57509
Z = 0.00704
Laboratory for Manufacturing Systems and Automation
Dr.Dimitris Mourtzis
A Lathe Example
P3 :
Χ = 0.745 + 0.020
Ζ = − ( 0.250 + ∆Κ )
Χ = 0.765
Ζ = − 0.25167
P4 :
Note: 0.25167 in Z formula from P3 Z calculation
Χ = 0.745 + 0.020( COS 37° )
Χ = 0.76097
[
Z = − 0.25167 − 0.020( SI 37° )
]
Z = − 0.23963
Laboratory for Manufacturing Systems and Automation
Dr.Dimitris Mourtzis
Summary
Right – angle trigonometry is the mathematical science of
solving right triangles
The sine of an angle equals the side opposite the angle
divided by the hypotenuse of the triangle
The cosine of an angle equals the side adjacent to the angle
divided by the hypotenuse of the triangle
The tangent of an angle equals the side opposite the angle
divided by the side adjacent to the angle
The use of trigonometry is necessary for determining cutter
offsets for linear and circular interpolation and for
determining other part information from a blueprint
Laboratory for Manufacturing Systems and Automation
Dr.Dimitris Mourtzis
Vocabulary Introduced in this chapter
Cosine
Cutter offsets
Jig borer coordinates
Sine
Tangent
Trigonometry
Laboratory for Manufacturing Systems and Automation
Dr.Dimitris Mourtzis
Review Questions
FIGURE 88-11 Triangle for review question #2
Laboratory for Manufacturing Systems and Automation
Dr.Dimitris Mourtzis
Review Questions
FIGURE 88-12 Part drawing for review question #3
Laboratory for Manufacturing Systems and Automation
Dr.Dimitris Mourtzis
Review Questions
FIGURE 88-13 Part drawing for review question #4
Laboratory for Manufacturing Systems and Automation
Dr.Dimitris Mourtzis